ECON 351* -- A Guide to Hypothesis Testing M.G. Abbott

A Guide to Hypothesis Testing in Linear Regression Models Tests of One Coefficient Restriction: One Restriction on One Coefficient H0 specifies only one equality restriction on one coefficient.

Two-Tail Tests of One Restriction on One Coefficient • Example: H0: βj = bj versus H1: βj ≠ bj where bj is a specified constant. • Use: either a two-tail t-test or an F-test. • Test Statistics:

tse

t N kjj j

j

( $ )$

$( $ )~ [ ]β

β β

β=

−− ⇒ sample value =

)ˆ(esbˆ

)ˆ(j

jjj0

β

−β=βt .

( )

FVar

F N kj

j j

j

( $ )$

$ ( $ )~ [ ,β

β β

β=

−−

2

1 ] ⇒ sample value = ( )

)ˆ(raVbˆ

)ˆ(j

2

jjj0 β

−β=βF .

Note: [ ] or t Fj j( $ ) ( $ )β β2= )ˆ(F)ˆ(t jj β=β and [ ] ]kN,1[F]kN[t 2

2/ −=− αα .

• Decision Rules:

Reject H0 if ]kN[tt 2/0 −> α or two-tail p-value for ( ) α<>= 00 ttPrt ; or p-value for ]kN,1[FF0 −> α α<>= )FFPr(F 00 .

Retain H0 if ]kN[tt 2/0 −≤ α or two-tail p-value for ( ) α≥>= 00 ttPrt ; or p-value for ]kN,1[FF0 −≤ α α≥>= )FFPr(F 00 .

ECON 351*: Guide to Hypothesis Testing Page 1 of 5 351guide.doc

ECON 351* -- A Guide to Hypothesis Testing M.G. Abbott

One-Tail Tests of One Restriction on One Coefficient • Examples: H0: βj = bj (or βj ≥ bj) versus H1: βj < bj a left-tail test H0: βj = bj (or βj ≤ bj) versus H1: βj > bj a right-tail test • Use: a one-tail t-test. • Test Statistic:

tse

t N kjj j

j

( $ )$

$( $ )~ [ ]β

β β

β=

−− ⇒ sample value =

)ˆ(es

bˆ)ˆ(t

j

jjj0

β

−β=β .

• Decision Rules -- left-tail t-test:

Reject H0 if or one-tail p-value for ]kN[tt 0 −−< α α<<= )ttPr(t 00 ; Retain H0 if or one-tail p-value for ]kN[tt 0 −−≥ α α≥<= )ttPr(t 00 .

• Decision Rules -- right-tail t-test:

Reject H0 if or one-tail p-value for ]kN[tt 0 −> α α<>= )ttPr(t 00 ; Retain H0 if or one-tail p-value for ]kN[tt 0 −≤ α α≥>= )ttPr(t 00 .

ECON 351*: Guide to Hypothesis Testing Page 2 of 5 351guide.doc

ECON 351* -- A Guide to Hypothesis Testing M.G. Abbott

Tests of One Linear Restriction on Two or More Coefficients H0 specifies only one linear restriction on two or more regression coefficients.

Two-Tail Tests of One Linear Restriction on Two Coefficients • Example: H0: cjβj + chβh = c0 versus H1: cjβj + chβh ≠ c0. • Use: either a two-tail t-test or an F-test. • Test Statistics:

( ) ]kN[t]kN[t~)ˆcˆc(es

)cc()ˆcˆc(ˆcˆct 1hhjj

hhjjhhjjhhjj −=−

β+β

β+β−β+β=β+β

where: )ˆcˆc(raV)ˆcˆc(e hhjjhhjj β+β=β+βs

. )ˆ,ˆ(voCcc2)ˆ(raVc)ˆ(raVc)ˆcˆc(raV hjhjh2hj

2jhhjj ββ+β+β=β+β

sample value = ( ))ˆcˆc(esc)ˆcˆc(ˆcˆct

hhjj

0hhjjhhjj0 β+β

−β+β=β+β .

( ) [ ]]kN,1[F]kN,1[F~

)ˆcˆc(raV)cc()ˆcˆc(ˆcˆcF 1

hhjj

2

hhjjhhjjhhjj −=−

β+β

β+β−β+β=β+β

where: . )ˆ,ˆ(voCcc2)ˆ(raVc)ˆ(raVc)ˆcˆc(raV hjhjh2hj

2jhhjj ββ+β+β=β+β

sample value = ( ) [ ])ˆcˆc(raV

c)ˆcˆc(ˆcˆcFhhjj

2

0hhjjhhjj0 β+β

−β+β=β+β

• Decision Rules:

Reject H0 if ]kN[tt 2/0 −> α or two-tail p-value for ( ) α<>= 00 ttPrt ; or p-value for ]kN,1[FF0 −> α α<>= )FFPr(F 00 .

Retain H0 if ]kN[tt 2/0 −≤ α or two-tail p-value for ( ) α≥>= 00 ttPrt ; ] or p-value for kN,1[FF0 −≤ α α≥>= )FFPr(F 00 .

ECON 351*: Guide to Hypothesis Testing Page 3 of 5 351guide.doc

ECON 351* -- A Guide to Hypothesis Testing M.G. Abbott

One-Tail Tests of One Linear Restriction on Two Coefficients • Examples: H0: cjβj + chβh = c0 vs. H1: cjβj + chβh < c0 a left-tail test H0: cjβj + chβh = c0 vs. H1: cjβj + chβh > c0 a right-tail test • Use: a one-tail t-test. • Test Statistic:

( ) ]kN[t]kN[t~)ˆcˆc(es

)cc()ˆcˆc(ˆcˆct 1hhjj

hhjjhhjjhhjj −=−

β+β

β+β−β+β=β+β

where )ˆcˆc(raV)ˆcˆc(e hhjjhhjj β+β=β+βs

)ˆ,ˆ(voCcc2)ˆ(raVc)ˆ(raVc)ˆcˆc(raV hjhjh2hj

2jhhjj ββ+β+β=β+β .

sample value = ( ))ˆcˆc(esc)ˆcˆc(ˆcˆct

hhjj

0hhjjhhjj0 β+β

−β+β=β+β .

• Decision Rules -- left-tail t-test:

Reject H0 if or one-tail p-value for ]kN[tt 0 −−< α α<<= )ttPr(t 00 ; Retain H0 if or one-tail p-value for ]kN[tt 0 −−≥ α α≥<= )ttPr(t 00 .

• Decision Rules -- right-tail t-test:

Reject H0 if or one-tail p-value for ]kN[tt 0 −> α α<>= )ttPr(t 00 ;

Retain H0 if or one-tail p-value for ]kN[tt 0 −≤ α α≥>= )ttPr(t 00 .

ECON 351*: Guide to Hypothesis Testing Page 4 of 5 351guide.doc

ECON 351* -- A Guide to Hypothesis Testing M.G. Abbott

ECON 351*: Guide to Hypothesis Testing Page 5 of 5 351guide.doc

Tests of Two or More Linear Coefficient Restrictions H0 specifies two or more linear coefficient restrictions. • Example: H0: β2 = β4 and β3 = β5

H1: β2 ≠ β4 and/or β3 ≠ β5 • Use: a general F-test; only an F-test can be used to test jointly two or more

coefficient restrictions. • Test Statistics: Either of the following two general F-statistics.

)kN(RSS

)kk()RSSRSS(dfRSS

)dfdf()RSSRSS(F

1

010

11

1010

−−−

=−−

= .

)kN()R1(

)kk()RR(df)R1(

)dfdf()RR(F 2

U

02R

2U

12U

102R

2U

−−−−

=−

−−= .

Null distribution: ]kN,kk[F]df,dfdf[F~F 0110 −−=− .

• Sample value of F-statistic under H0 = F0. • Decision Rules:

Reject H0 if ]kN,kk[F]df,dfdf[FF 01100 −−=−> αα or p-value for α<>= )FFPr(F 00 .

Retain H0 if ]kN,kk[F]df,dfdf[FF 01100 −−=−≤ αα or

p-value for α≥>= )FFPr(F 00 .